Summary
In this lesson, students will discover how a given side, side, and angle could result in 0, 1, or 2 triangles. They will then learn the Law of Sines and use it to find missing side lengths and angle measures. This lesson is intended to be taught before the Law of Cosines lesson and after students learn about inverse trigonometric functions.
Essential Question(s)
How can indirect measurements help calculate unknown distances and angle measurements?
Snapshot
Engage
Students use the Not Like the Others strategy to recall triangle vocabulary.
Explore
Students explore the ambiguous case of side-side-angle using GeoGebra.
Explain
Students formalize their understanding of the ambiguous case and the Law of Sines.
Extend
Students apply what they have learned to solve the unknown side length within a puzzle.
Evaluate
Students reflect on their learning using the Stoplight Stickies strategy.
Materials
Lesson Slides (attached)
How Many Triangles handout (attached; one per student; printed front/back)
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Model Notes) document (attached; for teacher use)
Using Law of SSSSines handout (attached; one per student; printed front only)
Using Law of SSSSines (Sample Response) document (attached; for teacher use)
Pencils
Paper
Sticky notes (preferably red, yellow, and green; 1 sticky note per student)
Scientific calculator
Student devices with internet access
Engage
10 Minute(s)
Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s Essential Question. Slide 4 identifies the lesson’s Learning Objectives. Review each of these with your class to the extent you feel necessary.
Show slide 5 and introduce students to the Not Like the Others strategy. Instruct students to decide which given triangle is not like the others. Then ask for a few volunteers to share their reasoning. If time allows, ask for students who have different answers to share or ask the class if anyone can find an answer that someone has not yet heard.
Explore
20 Minute(s)
Display slide 6 and introduce students to the Think-Pair-Share strategy. Explain that students are going to start the GeoGebra activity individually as the “Think” portion of the strategy. Provide students with the link to the GeoGebra activity: https://www.geogebra.org/m/rsw7dspt. This interactive GeoGebra activity gives students the opportunity to see why certain given side-side-angle combinations result in 0, 1, or 2 triangles. It includes two applets: The first has a given acute angle, while the second has a given obtuse angle.
After giving students a couple of minutes to explore the GeoGebra activity, pass out the attached How Many Triangles handouts to each student. Direct their attention to the front of the handout: Acute Angle and the first applet: SSA Applet – Acute Angle. Remind them that they are to be individually thinking and completing their table.
Instruct students to find a partner or assign partners. Have pairs compare their results and confirm or revise each other’s noticed patterns.
Display slide 7 and ask for volunteers to share their generalized observations. Write these on the board so that the class can see the list. Continue until all patterns have been shared. Ask the class to compare what is on the board with their list and if anything is on the board that they did not record to check the statement. If students find counterexamples, remove or revise the claims accordingly.
Now direct students’ attention to the back of the How Many Triangles handout: Right or Obtuse Angle and the second applet: SSA Applet – Right or Obtuse Angle. Ask students to work in pairs to complete the table and record any observed patterns. This applet only uses an obtuse angle, but the observed patterns apply to both right and obtuse angles.
Display slide 8 and ask for volunteers to share their generalized observations. Write these on the board so that the class can see the list. Continue until all patterns have been shared. Ask the class to compare what is on the board with their list and if anything is on the board that they did not record to check the statement. If students find counterexamples, remove or revise the claims accordingly.
Make note of any misconceptions or concerns from the student responses and be sure to address them during the Explain portion of the lesson.
Explain
25 Minute(s)
Show slide 9 and tell the class that they now need to work together to write three statements that when given a side, side, and acute angle yield 0, 1, or 2 triangles (one statement each). Have students discuss with their partner what those three statements should be. As students are discussing, give each student a copy of the attached Guided Notes handout.
After a few minutes, if the class observed a relationship to the first leg length and 4 (or half of eight), ask the class the following questions to get them to see that 4 is unique to this triangle and that the 4 is 8·sin (30°) = (hypotenuse)·sin (θ), also known as the height of the right triangle.
What was significant about the triangle when the first leg was 4 units long?
What is the relationship between 4 and the other side?
Is there something that would relate 30° to one-half? Is there a trigonometric function that relates an angle, the side opposite of the angle, and the hypotenuse? We are going to be talking about the sine function today.
Is there another name we could use for that opposite side of the triangle?
Now that students have the vocabulary of “height” for the side length of 4, give students a few more minutes to think of their three statements about what would cause 0, 1, or 2 triangles.
Bring the class together for a whole-class discussion and determine what should actually be written in the notes.
Repeat this with two statements for a given side, side, and obtuse angle. Let students know that the patterns for obtuse angles also apply to a right angle.
Display slide 10 and complete the examples on the Guided Notes handout with the class.
Have students add their completed Guided Notes to their math notebook if that is a classroom norm.
Extend
20 Minute(s)
Instruct students to find a new partner, someone they have not yet worked with during this lesson or assign students new partners. Show slide 11 and pass out the Using Law of SSSSines handout.
Now is the time for students to apply what they have learned to determine the unknown side length.
Evaluate
5 Minute(s)
Display slide 12 and share the Stoplight Stickies strategy with the class and make available red, yellow, and green sticky notes to the students. Direct students to write a question or comment, on their sticky note, that represents their confidence level regarding their ability to use the Law of Sines. Guide students to where you want them to put their sticky notes.
If time allows, answer some of the questions on the red sticky notes to help resolve any confusion. Use students’ comments and questions to determine whether students need remediation or are ready for the next lesson: “Law of Cosines.”
Resources
K20 Center. (n.d.). Not Like the Others. Strategies. https://learn.k20center.ou.edu/strategy/77
K20 Center. (n.d.). Stoplight Stickies. Strategies. https://learn.k20center.ou.edu/strategy/92
K20 Center. (n.d.). Think-Pair-Share. Strategies. https://learn.k20center.ou.edu/strategy/139
Nerdstudy. (2017, February 11). Determining if Triangles Exist – Nerdstudy [Video]. YouTube. https://youtu.be/iNCFx3XR-hw
Nerdstudy. (2017, February 11). The Ambiguous Case for Sine Law – Nerdstudy [Video]. YouTube. https://youtu.be/RCyjglaJo5w